Question: All of the 3rd grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$46.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$10.00$ each for teachers and $$10.50$ each for students, and the group paid $$124.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 46.5}$ ${10x+10.5y = 124.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-10x-7y = -93}$ ${10x+10.5y = 124.5}$ Add the top and bottom equations together. $ 3.5y = 31.5 $ $ y = \dfrac{31.5}{3.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5x+3.5y = 46.5}$ to find $x$ ${5x + 3.5}{(9)}{= 46.5}$ $5x+31.5 = 46.5$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {10x+10.5y = 124.5}$ and get the same answer for $x$ ${10x + 10.5}{(9)}{= 124.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.